Question

Determine where the graph of \(f\) is below the graph of g by solving the inequality \(f(x) \leq g(x) .\) Graph \(f\) and g together. \(f(x)=x^{4}-4\) \(g(x)=3 x^{2}\)

Short answer

The graph of \(f(x)\) is below or equal to \(g(x)\) for \(-2 \leq x \leq 2\).

Step-by-step solution

- Set up the inequality

We need to determine where the graph of \(f\) is below the graph of \(g\). This is done by solving the inequality \(f(x) \leq g(x)\). Given \(f(x) = x^4 - 4\) and \(g(x) = 3x^2\), set up the inequality as follows:\[ x^4 - 4 \leq 3x^2 \]

- Rearrange the inequality

Move all terms to one side of the inequality to get a zero on the other side:\[ x^4 - 3x^2 - 4 \leq 0 \]

- Solve the inequality

To solve the inequality, find the roots of the corresponding equation\[ x^4 - 3x^2 - 4 = 0 \]Set \(y = x^2\), transforming the equation into a quadratic form:\[ y^2 - 3y - 4 = 0 \]Solve this quadratic equation using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):\[ y = \frac{3 \pm \sqrt{9 + 16}}{2} = \frac{3 \pm 5}{2} \]Thus,\[ y_1 = 4 \text{and}\ y_2 = -1 \]Since \(y = x^2\), we only consider non-negative root:\[ x^2 = 4 \rightarrow x = \pm 2 \]The inequality then becomes \[ (x^2 - 4)x^2 - 4 = 0 \rightarrow (x-2)(x+2)x(x+1)(x-1) \leq 0 \]

- Test intervals around the roots

The roots split the real number line into intervals. Test each interval to determine where the inequality holds:1. \(x \in (-\infty, -2)\): Test \(x = -3\): \(((-3)^2-4)((-3) + 2)((-3))(3+1) 25> 0 Konsequentsammelt 0\leq 0\)

- Graph the functions

Graph the functions \(f(x) = x^4 - 4\) and \(g(x) = 3x^2\) on the same set of axes. The solution to the inequality corresponds to the x-values where the graph of \(f(x)\) is below or touching the graph of \(g(x)\).

Key concepts

polynomial inequalities

Polynomial inequalities involve expressions with variables raised to different powers. They can be challenging because they often require factoring and testing intervals on a number line.
Here, we have the inequality:
\ x^4 - 4 \leq 3x^2 \
which we rearrange to:
\ x^4 - 3x^2 - 4 \leq 0 \
This is a polynomial inequality because it includes powers of x. To solve, we first find the roots by equating it to zero:
\ x^4 - 3x^2 - 4 = 0 \
By letting y=x^2, we transform this into a quadratic equation:
\ y^2 - 3y - 4 = 0 \
which simplifies to:
\ y = \frac{3 \pm \sqrt{9 + 16}}{2} \
giving us \(y_1 = 4 \) and \(y_2 = -1\). Since y = x^2 must be non-negative:
\ x^2 = 4 \rightarrow x = \pm 2 \
Testing points between these roots helps confirm where the inequality holds true.

graphing functions

Graphing is a powerful tool in understanding inequalities. By visualizing functions, we can see where one curve is above or below another. For this exercise, we need to graph:
\ f(x) = x^4 - 4 \
and
\ g(x) = 3x^2 \
Graphing these functions will help us see the regions where the inequality \(f(x) \leq g(x)\) is satisfied.
First, plot the key points and general shape of each function.
For \(f(x)\), it’s a quartic function centered at -4. It starts and ends at positive infinity for large |x|.
For \(g(x)\), a quadratic function that opens upwards starting from the origin.
By comparing these graphs, find the x-values where \(f(x)\) lies below or touches \(g(x)\).

quadratic equations

A key part of solving polynomial inequalities is turning them into quadratic equations when possible.
Here, we transformed
\ x^4 - 3x^2 - 4 = 0 \
into a quadratic equation by setting \(y = x^2\):(br> \ y^2 - 3y - 4 = 0 \
This step simplifies solving for roots, as quadratic equations are easier to handle.
The Quadratic Formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
was used, leading to roots \(y_1 = 4\ and \y_2 = -1\).
Only the non-negative root is considered because \(y = x^2\):
\ x^2 = 4 \implies x = \pm 2. \
This transforms our inequality into:
\ (x^2 - 4)x(x+1)(x-1) \leq 0. \
Understanding and solving quadratic equations is thus crucial for handling polynomial inequalities.